Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 3

Integral Calculus

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus—and the term integral can also refer to the notion of the anti-derivative. Using integral calculus, one can compute the area under an explicit function or approximate the area under highly nonlinear functions:

• Integral calculus is useful for pricing financial derivatives.
• The price of a derivatives contract is calculated as the present value of expected future payoffs that depend on the future asset price distribution.
• To deal with the non-normality features of asset return distributions, one has to use integral calculus to approximate the area under a skewed fat-tailed density function when computing option prices.
• Integral calculus is useful for Monte Carlo simulations that are widely used for pricing derivative instruments with option-type features.
• When pricing options with Monte Carlo simulation, it is necessary to generate a large sample of possible future asset prices that will produce possible future payoffs. To do so, one has to draw a large number of random variables from a specific distribution. From random number generator, one may have to rely on integral calculus depending on the choice of a probability distribution for underlying assets.

What you will learn after reading this chapter:

• The meaning of integration and its relationship to differentiation.
• What a Riemann integral is and ...